❞ كتاب VIBRATION OF LAMINATED COMPOSITE DECKS BEAMS ❝ ⏤ أسامة محمد المرضي سليمان

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In this study, the effects of both lamination
scheme and boundary conditions on the natural
frequencies of free vibration of laminated composite decks
beams were investigated. The problem is analyzed and
solved using the energy approach which is formulated by a
finite element model. Lamination schemes for symmetric
and non-symmetric laminated beams were studied. Six
boundary conditions are considered; clamped-free (CF),
hinged-hinged (HH), clamped-clamped (CC), hingedclamped
(HC), hinged-free (HF), free-free (FF). Each
beam has either movable ends or immovable ends. It is
found that both symmetrically and anti-symmetrically
laminated beams of similar size and end conditions have
equal natural frequencies which, generally, decrease as the
angle of orientation increases. Also, It is found that the
more constrained beams have the higher values of natural
frequencies of transverse vibration. However, the free-free
and hinged-free beams are found to have the highest
frequencies of transverse vibration amongst all beams
although they look less constrained. This behavior is due to
the fact that the first mode of the two beams is equal zero
(rigid body motion), and replaced by the second mode to
be the fundamental mode. The values of the natural
frequencies of longitudinal modes are found to be the same
for all beams with movable ends since they are generated
by longitudinal movements only. But for immovable ends,
the clamped-free and hinged-free beams have equal
frequencies in longitudinal vibration, and those of the
other beams are also the same.

International Journal of Engineering Applied Sciences and Technology, 2018
Vol. 3, Issue 1, ISSN No. 2455-2143, Pages 61-66
Published Online May 2018 in IJEAST (http://www.ijeast.com)
61
VIBRATION OF LAMINATED COMPOSITE
DECKS BEAMS
Osama Mohammed Elmardi Suleiman Imad-Eldin Mahmoud Mahdi
Department of Mechanical Engineering Department of Mechanical Engineering
Nile Valley University, Atbara, River Nile, Sudan Nile Valley University, Atbara, River Nile, Sudan
Abstract،X In this study, the effects of both lamination
scheme and boundary conditions on the natural
frequencies of free vibration of laminated composite decks
beams were investigated. The problem is analyzed and
solved using the energy approach which is formulated by a
finite element model. Lamination schemes for symmetric
and non-symmetric laminated beams were studied. Six
boundary conditions are considered; clamped-free (CF),
hinged-hinged (HH), clamped-clamped (CC), hingedclamped
(HC), hinged-free (HF), free-free (FF). Each
beam has either movable ends or immovable ends. It is
found that both symmetrically and anti-symmetrically
laminated beams of similar size and end conditions have
equal natural frequencies which, generally, decrease as the
angle of orientation increases. Also, It is found that the
more constrained beams have the higher values of natural
frequencies of transverse vibration. However, the free-free
and hinged-free beams are found to have the highest
frequencies of transverse vibration amongst all beams
although they look less constrained. This behavior is due to
the fact that the first mode of the two beams is equal zero
(rigid body motion), and replaced by the second mode to
be the fundamental mode. The values of the natural
frequencies of longitudinal modes are found to be the same
for all beams with movable ends since they are generated
by longitudinal movements only. But for immovable ends,
the clamped-free and hinged-free beams have equal
frequencies in longitudinal vibration, and those of the
other beams are also the same.
Keywords،X Finite Element Method, First Order Shear
Deformation Theory, Free Vibration, Laminated
Composites, Natural Frequencies, Deck beam.
I. INTRODUCTION
Laminar composites are those having alternating layers of
material bonded together in some manner and include thin
coatings, thicker protective surfaces, claddings, bimetallic,
laminates, and sandwiches. Laminated composite beams are
increasingly being used in many engineering applications in
the fields of mechanical and civil engineering, transportation
vehicles, marine, aviation and aerospace.
The papers, which are presented here as references, address
the problem of the free vibration of laminated composite
beams. A theoretical analysis of the vibration of composite
beams with solid cross sections was also presented by Teoh
and Huang [1], Chandrasekhara et al. [2], Abramovich [3]. In
those analyses, the equations of motion were based on a
Timoshenko beam model (shear deformation considered).
Numerical results showed the effect of the shear deformation
and fiber orientation on the natural frequencies. Again,
Abramovich and Livshits [4] presented exact solutions for the
free vibration of non-symmetrically laminated cross-ply
composite beams. Marur and Kant [5] and [6], and McCarthy
et al [7] applied higher order shear deformation theories to
solve the problem of the free vibration of composite beams.
The first-order shear deformation theory was used by Teboub
and Hajela [8] to analyze the free vibration of generally
layered composite beams. Hodges et al. [9] presented two
different methods, which were simple analytical method and
finite element method for the prediction of the natural
frequencies and mode shapes of composite beams. In addition
to the references mentioned above, references [10], [11], and
12] applied different techniques of the finite element method
for the same problem.
II. MATHEMATICAL FORMULATION
The time-dependent axial and transverse displacements fields
are:
ƒv ƒw ƒv ƒw ƒv ƒw
ƒv ƒw ƒv ƒw „ر
„ذ
„د
ƒ­
ƒ­ ƒy
W x z t w x t
U x z t u x t z x t
, , ,
, , , ƒض ,
(1)
Where, u and w are the axial and transverse displacements
at the mid-plane, z is the perpendicular distance from the
mid-plane to the layer plane, ƒض is the rotation of a plane after
deformation, and t is the time. The strain- displacement
relations are:
„آ „آ